Well-balanced quadruple systems

A set system ( V , F ) with uniform block size k is t-balanced if any t-subset of V is contained in lambda or lambda + 1 blocks of F for some nonnegative integer lambda. ( V , F ) is well-balanced if it is t-balanced for any positive integer t <= k. In this paper, we establish the necessary conditions for the existence of a t-balanced set system, which is closely related with packing numbers D lambda ( v , k , t ) and covering numbers C lambda ( v , k , t ). Then we concentrate on well-balanced quadruple systems (with block size k = 4). For t = 2 , 3, define the well-balanced packing number W D ( v , 4 , t ) to be the maximum number of blocks in any well-balanced t- ( v , 4 , 1 ) packing. Similarly, the well-balanced covering number W C ( v , 4 , t ) denotes the minimum number of blocks in any well-balanced t- ( v , 4 , 1 ) covering. The exact values of W D ( v , 4 , 2 ) and W C ( v , 4 , 2 ) are completely determined in this paper and the determinations of W D ( v , 4 , 3 ) and W C ( v , 4 , 3 ) are reduced to W C ( v , 4 , 3 ) where v equivalent to 6 , 7 ( mod 12 ).